CALIBRATION OF AN ELLIPSE S ALGEBRAIC EQUATION AND DIRECT DETERMINATION OF ITS PARAMETERS
|
|
- Julianna Hart
- 6 years ago
- Views:
Transcription
1 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 19 (003), CALIBRATION OF AN ELLIPSE S ALGEBRAIC EQUATION AND DIRECT DETERMINATION OF ITS PARAMETERS MOHAMED ALI SAID Abstract. The coefficients of an ellipse s algebraic equation are not unique. Multiplying these coefficients by a number δ 0 does not affect the ellipse s shape. In this paper it is shown that at a certain δ, called calibration number, direct relations between the coefficients and the parameters of the ellipse are found. This value of δ is found and is shown to be invariant. Useful results concerning invariants of an ellipse s equation are found using calibration. 1. Introduction 1.1. Motivation and Statement of the problem. Ellipses are of the most commonly used primitive models in applications fields like pattern recognition. Several aspects of ellipses still in active research see for example 3, 4,. The algebraic equation of an ellipse is given by (1) Ax + Bxy + Cy + Dx + Ey + F 0, B 4AC < 0 The coefficients A, B, C, D, E, and F provide little direct insight into the ellipse s shape. Moreover, these coefficients are not unique since they can be multiplied by a number δ 0 and still describe the same ellipse. So, extracting the parameters of an ellipse from the coefficients of its algebraic equation is an important issue. The five ellipse parameters are: the coordinates of its center x 0 and y 0, the lengths of the semi-major axis a and of the semi-minor axis b, and the angle θ between the x-axis and the major-axis. The ellipse s parametric equation 7 is: () x y x0 y 0 + cos θ sin θ sin θ cos θ So, the problems to be considered here are a cos t b sin t 0 t π i. To find an ellipse s parameters in terms of its algebraic equation coefficients, ii. To investigate the invariants of the ellipse s algebraic equation. 1.. Previous Work. In 5 the parameters are determined as follow: Find the ellipse s center (x 0, y 0 ) by solving the two equations: Ax 0 + (B/)y 0 D/ and (B/)x 0 + Cy 0 E/. Translate the coordinate system xy to x y so that the origin (0, 0) is translated to the center (x 0, y 0 ). The equation of the ellipse takes the form: Ax + Bx y + Cy + F 0. Find the angle θ using cot θ (A C)/B. Rotate the system x y to x y through the angle θ. The equation of the ellipse 000 Mathematics Subject Classification. 51M04. Key words and phrases. Calibration, Ellipse s parameters. 1
2 MOHAMED ALI SAID takes the form: A x + C y + F 0. Find the invariants of the ellipse s equation: (3) I 1 A + C, I A B/ B/ C, I A B/ D/ 3 B/ C E/ D/ E/ F the semi axes lengths are determined from: a I 3 /(I A ) and b I 3 /(I C ). In 6, the parameters are determined as follows: The angle θ is determined from cot θ (A C)/B. Rotate the coordinates system xy to x y through the angle θ. The algebraic equation of the ellipse takes the form A x + C y + D x + E y + F 0. Then, x 0 D /(A ), y 0 E /(C ), a G/A, and b G/C, where G F + G /(4A ) + E /(4C ). In 1 the parameters are determined as follows: Find the eigenvalues λ 1 and λ for the matrix A B/. B/ C Let ϕ(x, y) Ax +Bxy+Cy +Dx+Ey+F. Find the center (x 0, y 0 ) by solving the two equations ( ϕ/ x) x0,y 0 0 and ( ϕ/ y) x0,y 0 0. Find d ϕ(x 0, y 0 ). The lengths of the semi axes are a d/λ 1 and b d/λ. The angle θ is obtained by comparing the orthonormalized modal matrix with the standard rotation matrix. In the next section an alternative to the above methods is introduced and several useful subsidiary results are also provided.. The proposed method The following theorem introduces the calibration number δ and the calibrated algebraic equation of the ellipse, hence the ellipse s parameters are determined. Theorem. Let the algebraic equation of an ellipse be Ax + Bxy + Cy + Dx + Ey + F 0, where B 4AC < 0, then the parameters of that ellipse are obtained as follows: 1. Calibrate the equation by multiplying it by the calibration number δ, where then the calibrated equation will be and and δ 4 (CD + AE BDE) F (4AC B ) (4AC B ), Āx + Bxy + Cy + Dx + Ēy + F 0.. The lengths of the semi-major axis a and the semi-minor axis b are given by a Ā + C + (Ā C) + B b Ā + C (Ā C) + B. 3. The center (x 0, y 0 ) of the ellipse is given by x 0 y 0 BE CD 4AC B BD AE 4AC B BĒ C D 4Ā C B B D ĀĒ 4Ā C B.
3 CALIBRATION OF AN ELLIPSE S ALGEBRAIC EQUATION The angle θ between the x-axis and the major-axis of the ellipse is given by θ cot 1 (A C)/B cot 1 (Ā C)/ B. Proof. Eliminate the parameter t from equation () of the ellipse, the result is (x x0 ) cos θ + (y y 0 ) sin θ (x x0 )( sin θ) + (y y 0 ) cos θ (4) + 1 a b On expanding equation (4) and comparing the result with equation (1) we get (5) A a sin θ + b cos θ, B (b a ) sin θ cos θ, C a cos θ + b sin θ (6) (7) D x 0 A y 0 B, E y 0 C x 0 B F Ax 0 + Bx 0y 0 + Cy 0 a b solving equations (6) simultaneously for x 0 and y 0, then BE CD (8) x 0 4AC B, y BD AE 0 4AC B Replacing x 0 and y 0 in equation (7) by their values in equations (8), then (9) F CD + AE BDE 4AC B a b Using equations (5), the following is obtained (10) (11) 4AC B 4a b, A + C a + b A + B / + C a 4 + b 4, (A C) + B (a b ) (1) cot θ A C B Now, replace a b in equation (9) by its value in the first of equations (10), then (13) (4AC B ) + 4F (4AC B ) 4(CD + AE BDE) 0 Equation (13) is an important equation since it depends only on the coefficients of equation (1), the algebraic equation of the ellipse. If the coefficients of equation (1) satisfy equation (13), then equations (5 1) are valid. But if the coefficients of equation (1) do not satisfy equation (13) then equation (1) should be multiplied by a number δ 0 to force their coefficients to satisfy equation (13). This process is called calibration of the algebraic equation of the ellipse. Thus, to define the calibration number δ replace A, B, C, D, E, and F in equation (13) by δa, δb, δc, δd, δe, and δf respectively and solve for δ excluding δ 0, then: (14) δ 4 CD + AE BDE) F (4AC B ) (4AC B ) Now, let the calibrated equation be Āx + Bxy + Cy + Dx + Ēy + F 0, then in equations (5 1) the coefficients of the algebraic equation are replaced by the coefficients of the calibrated algebraic equation. Thus, the center (x 0, y 0 ) is obtained as in equations (8): BE CD BĒ C D (15) x 0 4AC B 4Ā C B, y BD AE 0 4AC B B D ĀĒ 4Ā C B Solving simultaneously equations (10) calibrated, the lengths of the semi-axes are: (16) a Ā + C + (Ā C) + B, b Ā + C (Ā C) + B
4 4 MOHAMED ALI SAID The angle θ between the major-axis and the x-axis is determined as in equation (1): (17) cot θ A C B The proof of the theorem is completed Ā C B It can be seen from equations (15 17) that only two parameters of the five parameters of the ellipse, namely the lengths of the semi axes, need the calibration of the equation. However, the calibration of the algebraic equation of the ellipse helps in deducing many useful relations like those in equations (10 1). Some variants of such relations are used as constraints in ellipse fitting method. For example in 4 the constraint used is 4AC B 1, also in other papers A+C 1, and A + B / + C 1 are used. The goodness of such constraints can now be seen in the light of the calibration process. Lemma 1. The invariants of the calibrated algebraic equation of an ellipse are: I 1 a + b, I a b, and I 3 a 4 b 4. Thus, the invariants are functions of the lengths of the semi axes and only two of them are independent, namely I 1 and I. Proof. I 1 Ā + C a + b, where use is made for the second in equations (10) calibrated. For I take account of equation (3) and the first of equations (10) calibrated, then I Ā C B /4 (4a b )/4 a b as required. For I 3 make use of equations (3), (5), and (6 7) calibrated, then I 3 F (4Ā C B ) (ĀĒ + C D B DĒ) a 4 b 4 (I ) 4 as required. This lemma agrees with the intuition that an ellipse s shape is completely determined by lengths of its semi axes regardless of its position in the plane. Lemma. For an ellipse s algebraic equation, the calibration number δ I 3 /(I ) and thus is invariant. Proof. Since I (4AC B )/4, and then δ 4 (CD + AE BDE) F (4AC B ) (4AC B ), I 3 F (4AC B ) (AE + CD BDE), 4 δ 4 4I 3 (4I ) I 3 I. Since I and I 3 are invariants then δ is invariant under the same coordinates transformations. Lemma 3. For the calibrated equation of an ellipse: i. Ā + B /+ C a 4 +b 4, and (Ā C) + B (a b ) both are invariants. ii. The eigenvalues of the matrix Ā B/ B/ C are λ 1 a and λ b. iii. a I 1 + (I 1 ) 4I / and b I 1 (I 1 ) 4I /.
5 CALIBRATION OF AN ELLIPSE S ALGEBRAIC EQUATION... 5 Proof. i. a 4 + b 4 (I 1 ) I and (a b ) a 4 + b 4 a b (I 1 ) I I ii. Simple. iii. Use the definition of I 1 and I in lemma 1 and solve for a and b. Special Cases. As mentioned before δ I 3 /(I ). If δ < 0 then I 3 > 0 and the ellipse is imaginary. While if δ 0, the ellipse degenerates to a point since I 3 0. Conclusion. The concept of calibration introduced here allows a deeper insight to ellipses and helps in solving some problems related to them. The concept of calibration can be extended directly to hyperbolas and almost the same results can be obtained. References 1 M. Afwat. Algebra and Analytic Geometry. Zagazig University, N. Bennet et al. A method to detect and characterize ellipses using hough transform. IEEE Trans. Pattern Anal. & Machine Intel., (4), April G. Farin. From conics to NURBS: A tutorial and survey. IEEE Computer Graphics & Applications, September A. Fitzgibbon et al. Direct least squares fitting of ellipses. IEEE Trans. Pattern Analysis & Machine Intelligence, 1(5), May V. A. Ilyin and E. G. Poznyak. Analytic Geometry. Mir Pub., pp D. F. Rogers. Mathematical Elements For Computer Graphics. McGraw-Hill, pp I. Zeid. CAD/CAM Theory and Practice. McGraw-Hill, pp Received September 0, 00.; in revised form April 7, 003. Dept, Faculty of Engineering, Zagazig University, Zagazig, Egypt address: mmamsaid@hotmail.com
arxiv: v1 [math.ho] 27 May 2017
arxiv:1705.09845v1 [math.ho] 7 May 017 The foci and rotation angle of an ellipse, E 0, as a function of the coefficients of an equation of E 0 Alan Horwitz 5/7/17 Abstract First, we give a formula for
More informationLinear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems.
Linear Algebra Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems May 1, 2018 () Linear Algebra May 1, 2018 1 / 8 Table of contents 1
More informationIntroduction to conic sections. Author: Eduard Ortega
Introduction to conic sections Author: Eduard Ortega 1 Introduction A conic is a two-dimensional figure created by the intersection of a plane and a right circular cone. All conics can be written in terms
More informationRotation of Axes. By: OpenStaxCollege
Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions,
More informationPure Math 30: Explained! 81
4 www.puremath30.com 81 Part I: General Form General Form of a Conic: Ax + Cy + Dx + Ey + F = 0 A & C are useful in finding out which conic is produced: A = C Circle AC > 0 Ellipse A or C = 0 Parabola
More informationSection 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION
Section 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION When you are done with your homework you should be able to Recognize, and apply properties of, symmetric matrices Recognize, and apply properties
More informationConic Sections and Polar Graphing Lab Part 1 - Circles
MAC 1114 Name Conic Sections and Polar Graphing Lab Part 1 - Circles 1. What is the standard equation for a circle with center at the origin and a radius of k? 3. Consider the circle x + y = 9. a. What
More informationSKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.
SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or
More information1. Projective geometry
1. Projective geometry Homogeneous representation of points and lines in D space D projective space Points at infinity and the line at infinity Conics and dual conics Projective transformation Hierarchy
More information8.6 Translate and Classify Conic Sections
8.6 Translate and Classify Conic Sections Where are the symmetric lines of conic sections? What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a
More informationSome Highlights along a Path to Elliptic Curves
11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational
More informationPrecalculus Table of Contents Unit 1 : Algebra Review Lesson 1: (For worksheet #1) Factoring Review Factoring Using the Distributive Laws Factoring
Unit 1 : Algebra Review Factoring Review Factoring Using the Distributive Laws Factoring Trinomials Factoring the Difference of Two Squares Factoring Perfect Square Trinomials Factoring the Sum and Difference
More informationChapter 1 Analytic geometry in the plane
3110 General Mathematics 1 31 10 General Mathematics For the students from Pharmaceutical Faculty 1/004 Instructor: Dr Wattana Toutip (ดร.ว ฒนา เถาว ท พย ) Chapter 1 Analytic geometry in the plane Overview:
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationx = x y and y = x + y.
8. Conic sections We can use Legendre s theorem, (7.1), to characterise all rational solutions of the general quadratic equation in two variables ax 2 + bxy + cy 2 + dx + ey + ef 0, where a, b, c, d, e
More informationCircles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).
Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.
More informationDistance and Midpoint Formula 7.1
Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units
More informationy d y b x a x b Fundamentals of Engineering Review Fundamentals of Engineering Review 1 d x y Introduction - Algebra Cartesian Coordinates
Fundamentals of Engineering Review RICHARD L. JONES FE MATH REVIEW ALGEBRA AND TRIG 8//00 Introduction - Algebra Cartesian Coordinates Lines and Linear Equations Quadratics Logs and exponents Inequalities
More informationIsometric Invariants of Conics in the Isotropic Plane Classification of Conics
Journal for Geometry and Graphics Volume 6 (2002), No. 1, 17 26. Isometric Invariants of Conics in the Isotropic Plane Classification of Conics Jelena Beban-Brkić Faculty of Geodesy, University of Zagreb,
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationA. Correct! These are the corresponding rectangular coordinates.
Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These
More informationHomework 2. Solutions T =
Homework. s Let {e x, e y, e z } be an orthonormal basis in E. Consider the following ordered triples: a) {e x, e x + e y, 5e z }, b) {e y, e x, 5e z }, c) {e y, e x, e z }, d) {e y, e x, 5e z }, e) {
More informationTEKS Clarification Document. Mathematics Algebra
TEKS Clarification Document Mathematics Algebra 2 2012 2013 111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12. Source: The provisions of this 111.31 adopted to
More information3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone
3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong
More informationcomplex dot product x, y =
MODULE 11 Topics: Hermitian and symmetric matrices Setting: A is an n n real or complex matrix defined on C n with the complex dot product x, y = Notation: A = A T, i.e., a ij = a ji. We know from Module
More informationSome examples of two-dimensional regular rings
Bull. Math. Soc. Sci. Math. Roumanie Tome 57(105) No. 3, 2014, 271 277 Some examples of two-dimensional regular rings by 1 Tiberiu Dumitrescu and 2 Cristodor Ionescu Abstract Let B be a ring and A = B[X,
More informationHonors Precalculus Chapter 8 Summary Conic Sections- Parabola
Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Definition: Focal length: y- axis P(x, y) Focal chord: focus Vertex x-axis directrix Focal width/ Latus Rectum: Derivation of equation of parabola:
More informationSESSION CLASS-XI SUBJECT : MATHEMATICS FIRST TERM
TERMWISE SYLLABUS SESSION-2018-19 CLASS-XI SUBJECT : MATHEMATICS MONTH July, 2018 to September 2018 CONTENTS FIRST TERM Unit-1: Sets and Functions 1. Sets Sets and their representations. Empty set. Finite
More informationGeometric meanings of the parameters on rational conic segments
Science in China Ser. A Mathematics 005 Vol.48 No.9 09 09 Geometric meanings of the parameters on rational conic segments HU Qianqian & WANG Guojin Department of Mathematics, Zhejiang University, Hangzhou
More informationFundamentals of Engineering (FE) Exam Mathematics Review
Fundamentals of Engineering (FE) Exam Mathematics Review Dr. Garey Fox Professor and Buchanan Endowed Chair Biosystems and Agricultural Engineering October 16, 2014 Reference Material from FE Review Instructor
More informationLinear Algebra, Summer 2011, pt. 3
Linear Algebra, Summer 011, pt. 3 September 0, 011 Contents 1 Orthogonality. 1 1.1 The length of a vector....................... 1. Orthogonal vectors......................... 3 1.3 Orthogonal Subspaces.......................
More information49. Green s Theorem. The following table will help you plan your calculation accordingly. C is a simple closed loop 0 Use Green s Theorem
49. Green s Theorem Let F(x, y) = M(x, y), N(x, y) be a vector field in, and suppose is a path that starts and ends at the same point such that it does not cross itself. Such a path is called a simple
More informationStandard Form of Conics
When we teach linear equations in Algebra1, we teach the simplest linear function (the mother function) as y = x. We then usually lead to the understanding of the effects of the slope and the y-intercept
More informationQuadratics. Shawn Godin. Cairine Wilson S.S Orleans, ON October 14, 2017
Quadratics Shawn Godin Cairine Wilson S.S Orleans, ON Shawn.Godin@ocdsb.ca October 14, 2017 Shawn Godin (Cairine Wilson S.S.) Quadratics October 14, 2017 1 / 110 Binary Quadratic Form A form is a homogeneous
More informationClassification of cubics up to affine transformations
Classification of cubics up to affine transformations Mehdi Nadjafikah and Ahmad-Reza Forough Abstract. Classification of cubics (that is, third order planar curves in the R 2 ) up to certain transformations
More informationMATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections
MATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections The aim of this project is to introduce you to an area of geometry known as the theory of conic sections, which is one of the most famous
More informationAmerican Computer Science League
1. ACSL THE JUNIOR VAMPIRE SLAYER 5 POINTS IF THIS PROGRAM IS RUN AT THE END OF THE CONTEST, IT HAS A 10 MINUTE TIME LIMIT 1. 1395 1. 15,93 2. 2187 2. 27,81 3. 386415 3. 465, 831 4. 2160 4. NONE 5. 4322
More informationFFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations
FFTs in Graphics and Vision Homogenous Polynomials and Irreducible Representations 1 Outline The 2π Term in Assignment 1 Homogenous Polynomials Representations of Functions on the Unit-Circle Sub-Representations
More informationRecall, to graph a conic function, you want it in the form parabola: (x x 0 ) 2 =4p(y y 0 ) or (y y 0 ) 2 =4p(x x 0 ), x x. a 2 x x 0.
Warm up Recall, to graph a conic function, you want it in the form parabola: (x x 0 ) =4p(y y 0 ) or (y y 0 ) =4p(x x 0 ), x x ellipse: 0 a + y y 0 b =, x x hyperbola: 0 y y 0 a b =or y y 0 x x 0 a b =
More informationPrecalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix
PICTURE: Parabolas Name Hr Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix Using what you know about transformations, label the purpose of each constant: y a x h 2 k It is common
More informationA A x i x j i j (i, j) (j, i) Let. Compute the value of for and
7.2 - Quadratic Forms quadratic form on is a function defined on whose value at a vector in can be computed by an expression of the form, where is an symmetric matrix. The matrix R n Q R n x R n Q(x) =
More informationCalculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science
Calculus III George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 251 George Voutsadakis (LSSU) Calculus III January 2016 1 / 76 Outline 1 Parametric Equations,
More informationOn the Normal Parameterization of Curves and Surfaces
On the Normal Parameterization of Curves and Surfaces Xiao-Shan Gao Institute of Systems Science, Academia Sinica, Beijing Shang-Ching Chou The University of Texas at Austin, Austin, Texas 78712 USA International
More informationThings You Should Know Coming Into Calc I
Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real
More informationClassification of algebraic surfaces up to fourth degree. Vinogradova Anna
Classification of algebraic surfaces up to fourth degree Vinogradova Anna Bases of algebraic surfaces Degree of the algebraic equation Quantity of factors The algebraic surfaces are described by algebraic
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Basic Information Instructor: Professor Ron Sahoo Office: NS 218 Tel: (502) 852-2731 Fax: (502)
More informationAdditivity Of Jordan (Triple) Derivations On Rings
Fayetteville State University DigitalCommons@Fayetteville State University Math and Computer Science Working Papers College of Arts and Sciences 2-1-2011 Additivity Of Jordan (Triple) Derivations On Rings
More informationCITY UNIVERSITY LONDON. BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION. ENGINEERING MATHEMATICS 2 (resit) EX2003
No: CITY UNIVERSITY LONDON BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2003 Date: August 2004 Time: 3 hours Attempt Five out of EIGHT questions
More informationConic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.
Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)
More informationUncertainties & Error Analysis Tutorial
Uncertainties & Error Analysis Tutorial Physics 118/198/1 Reporting Measurements Uncertainties & Error Analysis Tutorial When we report a measured value of some parameter, X, we write it as X X best ±
More informationELLIPTIC CURVES BJORN POONEN
ELLIPTIC CURVES BJORN POONEN 1. Introduction The theme of this lecture is to show how geometry can be used to understand the rational number solutions to a polynomial equation. We will illustrate this
More informationax 2 + bx + c = 0 where
Chapter P Prerequisites Section P.1 Real Numbers Real numbers The set of numbers formed by joining the set of rational numbers and the set of irrational numbers. Real number line A line used to graphically
More informationPrentice Hall Mathematics, Algebra Correlated to: Achieve American Diploma Project Algebra II End-of-Course Exam Content Standards
Core: Operations on Numbers and Expressions Priority: 15% Successful students will be able to perform operations with rational, real, and complex numbers, using both numeric and algebraic expressions,
More informationInstitutionen för matematik, KTH.
Institutionen för matematik, KTH. Contents 4 Curves in the projective plane 1 4.1 Lines................................ 1 4.1.3 The dual projective plane (P 2 )............. 2 4.1.5 Automorphisms of P
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 26 (2010), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 6 (010), 359 375 www.emis.de/journals ISSN 1786-0091 EXAMPLES AND NOTES ON GENERALIZED CONICS AND THEIR APPLICATIONS Á NAGY AND CS. VINCZE Abstract.
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationAlg Review/Eq & Ineq (50 topics, due on 01/19/2016)
Course Name: MAC 1140 Spring 16 Course Code: XQWHD-P4TU6 ALEKS Course: PreCalculus Instructor: Van De Car Course Dates: Begin: 01/11/2016 End: 05/01/2016 Course Content: 307 topics Textbook: Coburn: Precalculus,
More informationTo Be (a Circle) or Not to Be?
To Be (a Circle) or Not to Be? Hassan Boualem and Robert Brouzet Hassan Boualem (hassan.boualem@univ-montp.fr) received his Ph.D. in mathematics from the University of Montpellier in 99, where he studied
More informationOn the number of representations of n by ax 2 + by(y 1)/2, ax 2 + by(3y 1)/2 and ax(x 1)/2 + by(3y 1)/2
ACTA ARITHMETICA 1471 011 On the number of representations of n by ax + byy 1/, ax + by3y 1/ and axx 1/ + by3y 1/ by Zhi-Hong Sun Huaian 1 Introduction For 3, 4,, the -gonal numbers are given by p n n
More information1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...
Contents 1 Geometry of R 1.1 Conic Sections............................................ 1. Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................
More informationIntroduction. Chapter Points, Vectors and Coordinate Systems
Chapter 1 Introduction Computer aided geometric design (CAGD) concerns itself with the mathematical description of shape for use in computer graphics, manufacturing, or analysis. It draws upon the fields
More informationOn The Generalized Hohmann Transfer with Plane Change Using Energy Concepts Part I
Mechanics and Mechanical Engineering Vol. 15, No. (011) 183 191 c Technical University of Lodz On The Generalized Hohmann Transfer with Plane Change Using Energy Concepts Part I Osman M. Kamel Astronomy
More informationChapter 8. Orbits. 8.1 Conics
Chapter 8 Orbits 8.1 Conics Conic sections first studied in the abstract by the Greeks are the curves formed by the intersection of a plane with a cone. Ignoring degenerate cases (such as a point, or pairs
More informationMatrices A brief introduction
Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Matrices Semester 1, 2014-15 1 / 44 Definitions Definition A matrix is a set of N real or complex
More informationav 1 x 2 + 4y 2 + xy + 4z 2 = 16.
74 85 Eigenanalysis The subject of eigenanalysis seeks to find a coordinate system, in which the solution to an applied problem has a simple expression Therefore, eigenanalysis might be called the method
More informationHomework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.
Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one
More informationInfinite Matrices and Almost Convergence
Filomat 29:6 (205), 83 88 DOI 0.2298/FIL50683G Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Infinite Matrices and Almost Convergence
More informationKepler s Quartic Curve as a Model of Planetary Orbits
International Mathematical Forum, 3, 2008, no. 38, 1871-1877 Kepler s Quartic Curve as a Model of Planetary Orbits Takuma Kimura Grduate School of Science and Technology Hirosaki University, Hirosaki 036-8561,
More informationA Diophantine System Concerning Sums of Cubes
1 2 3 47 6 23 11 Journal of Integer Sequences Vol. 16 (2013) Article 13.7.8 A Diophantine System Concerning Sums of Cubes Zhi Ren Mission San Jose High School 41717 Palm Avenue Fremont CA 94539 USA renzhistc69@163.com
More informationALGEBRAIC LONG DIVISION
QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More informationSection 6.5. Least Squares Problems
Section 6.5 Least Squares Problems Motivation We now are in a position to solve the motivating problem of this third part of the course: Problem Suppose that Ax = b does not have a solution. What is the
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 18 (2002),
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 18 (2002), 77 84 www.emis.de/journals STRUCTURE REPRESENTATION IN OBJECT ORIENTED KNOWLEDGE REPRESENTATION SYSTEMS KATALIN BOGNÁR Abstract. This
More informationAldine I.S.D. Benchmark Targets/ Algebra 2 SUMMER 2004
ASSURANCES: By the end of Algebra 2, the student will be able to: 1. Solve systems of equations or inequalities in two or more variables. 2. Graph rational functions and solve rational equations and inequalities.
More informationME 301 THEORY OFD MACHİNES I SOLVED PROBLEM SET 2
ME 30 THEOY OFD MACHİNES I SOLVED POBLEM SET 2 POBLEM. For the given mechanisms, find the degree of freedom F. Show the link numbers on the figures. Number the links and label the joints on the figure.
More informationA Study of Kruppa s Equation for Camera Self-calibration
Proceedings of the International Conference of Machine Vision and Machine Learning Prague, Czech Republic, August 14-15, 2014 Paper No. 57 A Study of Kruppa s Equation for Camera Self-calibration Luh Prapitasari,
More informationHi AP AB Calculus Class of :
Hi AP AB Calculus Class of 2017 2018: In order to complete the syllabus that the College Board requires and to have sufficient time to review and practice for the exam, I am asking you to do a (mandatory)
More informationBilinear and quadratic forms
Bilinear and quadratic forms Zdeněk Dvořák April 8, 015 1 Bilinear forms Definition 1. Let V be a vector space over a field F. A function b : V V F is a bilinear form if b(u + v, w) = b(u, w) + b(v, w)
More informationORTHONORMAL SAMPLING FUNCTIONS
ORTHONORMAL SAMPLING FUNCTIONS N. KAIBLINGER AND W. R. MADYCH Abstract. We investigate functions φ(x) whose translates {φ(x k)}, where k runs through the integer lattice Z, provide a system of orthonormal
More informationVideo 15: PDE Classification: Elliptic, Parabolic and Hyperbolic March Equations 11, / 20
Video 15: : Elliptic, Parabolic and Hyperbolic Equations March 11, 2015 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, 2015 1 / 20 Table of contents 1 Video 15: : Elliptic, Parabolic
More informationIntroduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves
Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves 7.1 Ellipse An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r from two fixed
More informationMATHEMATICS SYLLABUS SECONDARY 4th YEAR
European Schools Office of the Secretary-General Pedagogical Development Unit Ref.:010-D-591-en- Orig.: EN MATHEMATICS SYLLABUS SECONDARY 4th YEAR 6 period/week course APPROVED BY THE JOINT TEACHING COMMITTEE
More informationFACTORISING ALL TYPES. Junior Cert Revision
FACTORISING ALL TYPES Junior Cert Revision 2017 JCHL Paper 1 Question 12 (a) Factorise n 2 11n + 18. n 2 11n + 18 n 9 n 2 n 2 11n + 18 n 9 n 2 9n 2n 18 9 2 6 3 18 1 2n 9n 11n 2017 JCHL Paper 1 Question
More informationNotes from the Marking Centre - Mathematics Extension 2
Notes from the Marking Centre - Mathematics Extension Question (a)(i) This question was attempted well, with most candidates able to calculate the modulus and argument of the complex number. neglecting
More information[3] (b) Find a reduced row-echelon matrix row-equivalent to ,1 2 2
MATH Key for sample nal exam, August 998 []. (a) Dene the term \reduced row-echelon matrix". A matrix is reduced row-echelon if the following conditions are satised. every zero row lies below every nonzero
More informationVarsity Meet 3 January 8, 2014 Round 1: Similarity and Pythagorean Theorem
Round 1: Similarity and Pythagorean Theorem 1. A baseball diamond is a square 90 feet on each side. The right fielder picks up the ball 30 feet behind first base (at point R) and throws it to the third
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationChapter 11 Parametric Equations, Polar Curves, and Conic Sections
Chapter 11 Parametric Equations, Polar Curves, and Conic Sections ü 11.1 Parametric Equations Students should read Sections 11.1-11. of Rogawski's Calculus [1] for a detailed discussion of the material
More informationApplied Linear Algebra
Applied Linear Algebra Guido Herweyers KHBO, Faculty Industrial Sciences and echnology,oostende, Belgium guido.herweyers@khbo.be Katholieke Universiteit Leuven guido.herweyers@wis.kuleuven.be Abstract
More informationMATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane
MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More informationAlgebra II Final Exam Semester II Practice Test
Name: Class: Date: Algebra II Final Exam Semester II Practice Test 1. (10 points) A bacteria population starts at,03 and decreases at about 15% per day. Write a function representing the number of bacteria
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationChapter 2. First-Order Differential Equations
Chapter 2 First-Order Differential Equations i Let M(x, y) + N(x, y) = 0 Some equations can be written in the form A(x) + B(y) = 0 DEFINITION 2.2. (Separable Equation) A first-order differential equation
More informationEdexcel GCE Further Pure Mathematics (FP1) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Further Pure Mathematics (FP1) Required Knowledge Information Sheet FP1 Formulae Given in Mathematical Formulae and Statistical Tables Booklet Summations o =1 2 = 1 + 12 + 1 6 o =1 3 = 1 64
More informationSenior Math Circles February 11, 2009 Conics II
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles February 11, 2009 Conics II Locus Problems The word locus is sometimes synonymous with
More information9.1 Circles and Parabolas. Copyright Cengage Learning. All rights reserved.
9.1 Circles and Parabolas Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in
More informationLecture 17. Implicit differentiation. Making y the subject: If xy =1,y= x 1 & dy. changed to the subject of y. Note: Example 1.
Implicit differentiation. Lecture 17 Making y the subject: If xy 1,y x 1 & dy dx x 2. But xy y 2 1 is harder to be changed to the subject of y. Note: d dx (f(y)) f (y) dy dx Example 1. Find dy dx given
More informationEcon Slides from Lecture 8
Econ 205 Sobel Econ 205 - Slides from Lecture 8 Joel Sobel September 1, 2010 Computational Facts 1. det AB = det BA = det A det B 2. If D is a diagonal matrix, then det D is equal to the product of its
More informationWeek Quadratic forms. Principal axes theorem. Text reference: this material corresponds to parts of sections 5.5, 8.2,
Math 051 W008 Margo Kondratieva Week 10-11 Quadratic forms Principal axes theorem Text reference: this material corresponds to parts of sections 55, 8, 83 89 Section 41 Motivation and introduction Consider
More information